Extended hard-sphere model and collisions of
cohesive particles
Pawel Kosinski, Alex C. Hoffmann
The University of Bergen, Department of Physics and Technology
Bergen, Norway
Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
Abstract
In two earlier papers the present authors modified a standard hard-sphere
particle/wall and particle/particle collision model to account for the presence of adhesive or cohesive interaction between the colliding particles:
the problem is of importance for modelling particle-fluid flow using the
Lagrangian approach. This technique, which involves a direct numerical
simulation of such flows, is gaining increasing popularity for simulating
e.g. dust transport, flows of nanofluids and grains in planetary rings.
The main objective of the previous papers was to formally extend the
impulse-based hard sphere model, while suggestions for quantifications of
the adhesive or cohesive interaction were made. This present paper gives
an improved quantification of the adhesive/cohesive interaction for use in
the extended hard-sphere model for cases where the surfaces of the colliding bodies are “dry”, e.g. there is no liquid-bridge formation between the
colliding bodies. This quantification is based on the Johnson-KendallRoberts (JKR) analysis of collision dynamics but includes, in addition,
dissipative forces using a soft-sphere modelling technique. In this way
the cohesive impulse, required for the hard-sphere model, is calculated
together with other parameters, namely the collision duration and the
restitution coefficient. Finally a dimensional analysis technique is applied
to fit an analytical expression to the results for the cohesive impulse that
can be used in the extended hard-sphere model. At the end of the paper
we show some simulation results in order to illustrate the model.
Keywords
fluid-particle flows; hard-sphere model; restitution coefficient; direct numerical simulation; cohesion; adhesion; agglomeration; dimensional analysis; Lagrangian approach; two-phase flows; collisions
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
1
Introduction
Eulerian-Lagrangian simulation of fluid-particle flows offers opportunities for
implementing the physical phenomena involved in the process directly, for example particle-particle or particle-wall collisions. Due to the computational intensity the scale of the process that can be simulated with this technique is very
limited, but the results can be used as input to Eulerian-Eulerian simulations
on a larger scale. This paper is focused on the modelling of particle-particle and
particle-wall collisions for use in Eulerian-Lagrangian simulations. This technique was also used in earlier papers by the present authors, e.g. [1–3] and it
has a wide range of applications: transport of particles, modelling of hydrate
flows, nanofluids, macrofluids or cosmic dusts in planetary rings.
Of the two types of technique for modelling collisions in fluid-particle flows,
namely the hard-sphere and the soft-sphere techniques, the former is by far
the faster, giving the post-collisional translational and angular velocity vectors
by direct calculation rather than having to solve differential equations to find
them. The soft-sphere technique is much slower rendering it impracticable in
many contexts, but has the advantage that it is easier to implement the physics
of the collision directly.
Roughly, the hard-sphere collision models are based on time-integrating
Newtons
equation of motion for the colliding bodies to obtain impulses, J ≡
R
F dt, where F is the force acting on the colliding bodies, and formulating
the appropriate system of equations for the impulses and their moments (see
below) for the collision. In this way relations between the post-collisional and
pre-collisional translational and angular velocities can be found directly. The
hard-sphere model used in this work is based on a model probably presented for
the first time by Matsumoto and Saito [4], this model is also described in detail
in the textbook by Crowe, Tsuji and Sommerfeld [5] and is briefly accounted for
in recent papers by the present authors [6, 7]. This type of hard-sphere model
incorporates a restitution coefficient to account for an inelastic component in
the collisional particle deformation and a Coulombian friction factor to account
for sliding during the collision.
The hard sphere models in general and the one used here in particular, however, do not account for adhesion or cohesion between the colliding bodies. This
is an important drawback, since in many real processes cohesion, whether due
to van der Waals interaction, liquid bridging or even electrostatic or magnetic
forces, plays a crucial part in the development of the process as particles may
form agglomerates far larger and far fewer than the primary particles, and particles may deposit on walls or obstructions in the system. Cohesion and adhesion
are, in practice, particularly relevant e.g. in systems involving moist particles
or very fine particles, such as nanoparticles.
The focus of the two recent research papers by the present authors mentioned above was to overcome this problem. In [6,7] the hard-sphere model was
extended by including cohesion and adhesion into the model so that phenomena like deposition and agglomeration could be simulated directly. This was
done by re-writing the basic equations in the hard-sphere model including an
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
extra attractive force acting in the point of collision. Such an attractive force
gives rise to the particle or particles entering a potential well upon collision
and, depending on the strength of this well and the dissipative processes during
the collision robbing the particles of mechanical energy, the collision may result
in particle deposition on a wall or particle agglomeration. Thus, with proper
quantification of the attractive interaction, the extended model can distinguish
whether or not a given collision leads to particle deposition on a wall or particle
agglomeration, the classical model cannot lead to agglomeration except in the
limit of zero coefficient of restitution.
We mention here that Weber et al. [8] proposed a similar concept in making
use of a simple square-well potential to account for adhesion in hard-sphere
models. Saito et al. [9] studied the collisions of nanoclusters using molecular
dynamics. They found that reorientation of the plane of collision during the
collision could significantly influence the coefficient of restitution, causing it in
some cases to become negative (see also below).
Physically these additional forces may be of different types: the van der
Waals interaction, liquid bridging, electrostatic or magnetic forces. In references [6] and [7] the attractive forces were quantified assuming they were of the
van der Waals type. Due to the nature of the problems, namely the difference
in geometries between a collision of a particle with a flat wall on the one hand
and a collision between two particles on the other, the strategies for the quantifications were different in the two papers, although in both cases the analyses
lead to useful quantifications of the attractive force that can be used in simulations. van der Waals interaction was used as a basis, since this interaction
is always present (see for example [10–12]). Weber et al. [8] also used van der
Waals interaction when quantifying their square-well potential.
More specifically in [6], discussing particle-wall collisions, quantification of
the attractive impulse was based on the assumption that van der Waals interaction acts on the particle during approach to and departure from the wall only,
the duration of the contact itself being so short that it is not contributing to
the attractive impulse; this is in line with the method of Weber et al. [8]. In [7],
describing particle-particle collisions, this technique was not feasible due to the
geometry of the collision, and the assumption was made that the duration of
contact is finite (the duration being found using Hertz theory [13]) and that
van der Waals attraction acts during the period of contact only. The latter of
these two techniques could also be applied to particle-wall collisions, while, as
mentioned, the former is difficult to apply to particle-particle collisions.
This present work builds on the second of these two approaches, aiming to
improve on the quantification of the attractive impulse and implementing it into
both particle-wall and particle-particle collision simulations.
Since the impulse is a time-integral of the attractive force during the contact
period the duration of the contact is crucial. In this paper this duration will
be estimated using the theory of Johnson, Kendall and Roberts [14], or the
JKR-theory. This theory analyses two contacting (but not colliding) particles,
taking into account adhesion in the contact. In addition to the JKR theory
use will be made of the model developed in [15], wherein the dissipative force
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
during a collision was calculated. This strategy requires numerical integration
in time to find the cohesive impulse. Doing this, it will be possible to evaluate
the attractive impulse. This actually amounts to using the soft-sphere approach
(see e.g. the classical work [16] or the recent review [17]).
In order to incorporate the results of the above analysis in the hard-sphere
model an analytical expression for the attractive impulse must be derived from
the results of the analysis. This is done by carrying out a dimensional analysis
of the problem and fitting a relation between the resulting dimensionless groups
to the analysis results. The analysis shows that the attractive impulse depends
on: the initial relative speed of the colliding particles, their material properties (such as Young’s moduli, Poisson ratios and density), their size and the
adhesion/cohesion (described e.g. by the surface energy) acting between them.
Using Buckingam’s Π theorem it is possible to find the set of non-dimensional
groups and, running a series of numerical experiments, the required impulse can
be related to these groups in a sort of “empirical” relation that can be directly
used in the extended hard-sphere model.
This technique will also be shown to be useful for calculation of other quantities that constitute an input to the hard-sphere model, such as the restitution
coefficient.
2
A brief description of the extended hard-sphere
model
Since this paper is a further development for use in the previously published [6,7]
extended hard-sphere model, and the reader may wish some basic information
about this, the most important steps in the derivations of both the extended
particle-wall and particle-particle collision models are given below to make this
paper more self-contained.
2.1
Particle-wall collisions
The derivation of the standard hard-sphere particle-wall collision model [4, 5]
begins with dividing the process into two periods:
(a) compression (i.e. the particle material is deformed after the first contact)
and
(b) recovery (i.e. the elastic deformation is released).
In addition three cases that account for particle sliding during the collision
process are distinguished:
(I) the particle stops sliding during the compression period;
(II) the particle stops sliding in the recovery period;
(III) the particle continues to slide throughout the entire collision.
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
For all the cases a system of impulse equations are then written (see also Figure 1):
m∆v = J
(1)
I∆ω = rn × J,
where m is particle mass, r is particle radius, n is the unit normal vector to the
plane of impact, ∆v, ∆ω, J are vectors of, respectively, linear velocity change,
angular velocity change and impulse written for all the periods and cases.
y
z
x
Jy
J
Jt
Figure 1: Illustration of a particle-wall collision. In the extended model the
normal impulse, Jy = (0, Jy , 0), is the sum of the repulsive impulse due to the
material deformation and the cohesive impulse
For cases I and II, the above equations involve 27 unknowns, which are the
components of the vectors v, ω and J at the end of the compression, sliding and
recovery periods (the latter ones are the final result). There are a total of 18
relations between the unknowns in Equation (1). More relations are obtained
by considering the motion of the particle surface in the contact point, among
other things requiring that there is no tangential velocity of the particle surface
in the contact point at the end of the sliding period. Finally the rest of the
relations required to close the system of equations are obtained by invoking the
coefficient of restitution, e, and the Coulombic dynamic coefficient of friction, f .
Writing the impulse equations for Coulombic friction during the sliding period
introduces two new variables, the direction cosines of the sliding, and the final
result is therefore 29 equations in 29 unknowns. Case III is a bit simpler. The
derivations and solution are given in references [4] and [5].
The standard model does not account for adhesion, and can therefore not
describe that the particle may deposit on the wall as a result of the collision. The
present authors proposed that it is possible to include an adhesive force acting in
the point of contact between the particle and the wall. In the standard
R model,
the normal component of the impulse acting on the particle, Jy = Fy dt, is
away from the wall. In the extended model another force/impulse is included
acting in the opposite direction and accounting for adhesion.
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
This extension of the model led to a modified system of equations, having
the following solutions for Cases I and II:
5 (0) 2r (0) v − ωz
7 x
5
(2)
(1)
Jy,c Jy,c
+
= em − vy(0) −
m
m
5 (0) 2r (0) = vz + ωx
7
5
(2)
vz
=
r
= ωy(0)
vx(2) =
(2a)
vy(2)
(2b)
vz(2)
ωx(2)
ωy(2)
(2c)
(2d)
(2e)
(2)
ωz(2) = −
vx
,
r
(2f)
where indices (0) and (2) denote, respectively, the states before and after the
collision, vx , vy , vz , ωx , ωy , ωz are the components of the linear and angular
(1)
(2)
velocity of the particle, Jy,c and Jy,c are the impulsive forces responsible for
adhesion during the compression and the recovery period, respectively, and r
is the particle radius. The restitution coefficient, em , is in the extended model
defined as in the original model, namely as the normal impulse during the
recovery period divided by that during the compression period, neglecting the
adhesive impulse.
In these and the following equations, as indeed in the original derivations,
the attractive impulse has been included explicitly in the model solutions rather
than being summed with the normal impulse due to the material deformation.
The set of solutions for Case III are:
#
"
(1)
(2)
Jy,c
Jy,c
(2)
(0)
(0)
(3a)
vx = vx + εx f vy (1 + em ) + εx f (2 + em )
+
m
m
!
(1)
(2)
Jy,c
Jy,c
(0)
(2)
vy = em −vy −
+
(3b)
m
m
#
"
(2)
(1)
Jy,c
Jy,c
(2)
(0)
(0)
(3c)
+
vz = vz + εz f vy (1 + em ) + εz f (2 + em )
m
m
#
"
(1)
(2)
5
5
Jy,c
Jy,c
(2)
(0)
(0)
ωx = ωx − εz f vy (em + 1) − εz f (2 + em )
(3d)
+
2r
2r
m
m
ωy(2) = ωy(0)
ωz(2) = ωz(0) +
(3e)
"
(1)
(2)
5
5
Jy,c
Jy,c
εx f vy(0) (em + 1) + εx f (2 + em )
+
2r
2r
m
m
#
,
(3f)
where εx and εz are the direction cosines for the sliding motion and f is the
Coulombian friction coefficient.
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
Equations (2) and (3) reduce to the standard hard-sphere particle-wall collision model if the adhesive interaction is set to zero.
As before, an exact description of how to distinguish between the cases is
given in reference [6]. Here only the final results are given. Cases I and II occur
when:
q
2
(0)
(0) 2
(0)
(0) 2
m vx + aωz
+ vz − aωx
7
(1)
(2)
≤ [−mvy(0) (1 + em ) + em Jy,c
+ Jy,c
]f (4)
and in that case Eqs. (2) should be used. Otherwise, the collision is modelled
by the relations from Case III, i.e. described by Eqs. (3).
The relations above are valid for the case when the particle rebounds after
the collision. If [6]:
1
(1)
vy(0) <
(J (2) − em Jy,c
),
(5)
em m y,c
then the particle will deposit and its final velocities may be set to zero.
The mechanical restitution coefficient, em , is in the extended model defined as the restitution coefficient in the original model, namely as the normal
impulse during the recovery period divided by that during the compression period, neglecting the cohesive impulse. In other words em accounts only for the
”mechanical” properties of the particles, and therefore represents the collision
coefficient of restitution in the absence of cohesion.
The restitution coefficient, em , is not constant but depends on factors such as
initial relative speed, particle material etc (see e.g. Stronge [18]). The standard
hard-sphere model treats it, however, as input. In this paper, as shown later, it
is found by analysis of collision dynamics.
2.2
Particle-particle collisions
This section described the extended hard-sphere model for particle-particle collisions. As for particle-wall collisions the standard hard-sphere model is based
on writing impulse equations this time for two colliding particles (denoted by
subscripts 1 and 2, respectively):
m1 ∆v1 = J
m2 ∆v2 = −J
(6a)
(6b)
I1 ∆ω 1 = r1 n × J
I2 ∆ω 2 = r2 n × J.
(6c)
(6d)
Also for particle-particle collisions separate cases are distinguished. As discussed above, it turns out that for cases I and II in the particle-wall collision
model the same solutions are recovered, and in the particle-particle algorithm
only two cases are distinguished:
(i) the particles stop sliding during the collision or
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
(ii) they slide throughout the collision.
This simplifies the derivation of the model, at least in this respect. The particleparticle model reduces to the particle-wall model if the mass of one of the
particles is set to be “immovable” in the sense that its mass is set to infinite, its
the velocity is set to zero and the coordinate system is aligned with the plane
of impact with the normal axis pointing out of the immovable particle.
Having formulated additional relations between the unknowns involving the
relative motion of the particle surfaces in the point of impact and further relations invoking the restitution coefficient and the Coulombic frictions factor
in a similar manner as for particle-wall collisions, analytical solutions for the
post-collisional translational and angular velocities for both particles.
The extended hard-sphere model for particle-particle collisions [7] uses the
same approach as the particle-wall collision model: an additional force/impulse
describing any cohesive interaction is included in the model derivation.
This leads to the following solution of the system of equations giving the
post-collisional translational and angular velocities of both particles for case i:
Jn,c
m2
(n − f t) −
(1 + em )n · G(0) (n + f t)
m1
m1 + m2
Jn,c
m1
(0)
v2 = v2 −
(n − f t) +
(1 + em )n · G(0) (n + f t)
m2
m1 + m2
Jn,c
m2
5
(0)
(0)
(n × t)f −
−
(1 + em )n · G
ω1 = ω 1 +
2r1
m1
m1 + m2
5
Jn,c
m1
(0)
(0)
ω2 = ω 2 +
(n × t)f −
−
(1 + em )n · G
2r2
m2
m1 + m2
(0)
v1 = v1 +
(7a)
(7b)
(7c)
(7d)
and for case ii:
2
m2
Jn,c
m2
(0)
(0)
+
v1 =
n−
−
(1 + em )n · G
|G |t
m1
m1 + m2
7 m1 + m2 ct
(8a)
2
m1
m1
Jn,c
(0)
(0)
(0)
n−
v2 = v2 −
−
(1 + em )n · G
|G |t
m2
m1 + m2
7 m1 + m2 ct
(8b)
m
5
2
(0)
(0)
(n × t)|Gct |
(8c)
ω1 = ω 1 −
7r1
m1 + m2
m1
5
(0)
(0)
(n × t)|Gct |
.
(8d)
ω2 = ω 2 −
7r2
m1 + m2
(0)
v1
where n is the unit normal vector to the plane of impact pointing out of particle
1 (see Figure 2), while t is the tangential unit vector. In these equations G =
v1 − v2 is the relative velocity between the particle centres after the collision.
(0)
Gct is the relative velocity of the particles’ surfaces at the point of impact, just
before the impact, resolved in the direction tangential to the plane of impact:
(0)
(0)
(0)
Gct = G(0) − (G(0) · n)n + r1 ω1 × n + r2 ω 2 × n.
8
(9)
Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
Particle 2
n
Jt
P
t
J
Jn
Plane of!
collision
Particle 1
Figure 2: Illustration of a particle-particle collision. In the extended model the
normal impulse, Jn , is the sum of the repulsive impulse due to the material
deformation and the cohesive impulse
Also here the extended hard-sphere model differs from the standard one in
that a cohesive impulse Jn,c is included. Equations (7) and (8) reduce to the
standard model if Jn,c is set to zero.
The criterion distinguishing between the cases is:
n · G(0) < −
m1 + m2 Jn,c
1
2
(0)
+
|G |,
m1 m2 (1 + em ) 7 f (1 + em ) ct
(10)
which has to be fulfilled for Case ii to occur and in this case Jt will be given by:
Jt = −
2 m1 m2
(0)
|G |.
7 m1 + m2 ct
(11)
In this way the relations described by Eqs. 7 and 8 can be used directly
to find the new particle velocities. These formulae are valid for the case when
particles do not agglomerate (although the cohesive interaction does influence
their post-collisional velocities). Thus before using them it is necessary to check
whether the collision results in the particles agglomerating or not.
For Case i, agglomeration has taken place if the following conditions are
fulfilled:
m1 m2
em n · G(0)
(12)
Jn,c >
m1 + m2
and
|Jt | < |Jtl |,
(13)
where a typing error that occurred in the equivalent equation in [7] has been
corrected. In that equation, which is Eq. (20) in [7], the inequality sign should
be inverted, which is equivalent to taking the absolute values as done in Eq. (13)
in this paper, because Jt is always negative.
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
For Case i the condition described by Eq. (12) is sufficient.
An agglomerate (if formed) obtains a linear and angular velocity that can
be found by using the equations of conservation of momentum and moment of
momentum. The result is:
(0)
vagg =
(0)
m1 v1 + m2 v2
(m1 + m2 )
and
(0)
ωagg =
(14)
(0)
M1 + M2
,
Iagg
(15)
where: vagg and ω agg are the vectors of the linear and angular velocities of the
formed agglomerate, respectively, Iagg is the moment of inertia of the agglomer(0)
(0)
ate around its centre of mass and M1 and M2 are the moments of momenta
of the two colliding particles about the agglomerate centre of mass just before
the collision. These parameters can be found as:
Iagg =
and
2 2
m1 m2
(r1 + r2 )2
(r m1 + r22 m2 ) +
5 1
m1 + m2
(0)
(0)
(0)
M1 = (−r1 n) × m1 v1 + I1 ω 1
(0)
(0)
(0)
M2 = (r2 n) × m2 v2 + I2 ω2 ,
(16)
(17a)
(17b)
where: I1 and I2 are the particle moments of inertia around their own centres
(equal to I1 = (2/5)m1 r12 and I2 = (2/5)m2 r22 , respectively).
The details are given in ref. [7].
As mentioned above, Saitoh et al. [9] found that reorientation of the plane
of collision could influence the coefficient of restitution, and therefore the postcollisional velocities of colliding nanoclusters. In the present model reorientation
of the collision plane was taken into account in the agglomeration criterion (see
Figure 3 in [7], which is very similar to Figure 1 of Saitoh et al.). However, it
should be noted that the present model for the post-collisional velocities of the
separate particles does not account for this at present. The present model is
therefore accurate for collisions that are rather head-on, collisions where the cohesion is not strong enough to cause significant collision-plane reorientation and
collisions leading to agglomeration, but not for oblique collisions with significant
cohesion not leading to agglomeration.
For these simulations we have to make reservations about possible effects of
reorientation of the collision place in glancing collisions. Estimates, however,
of the extent of reorientation using the angular velocities of the particles while
in contact (see [7]), and the collision duration indicate that this effect is only
minor with the physical parameters used here, and in general if the particles are
not too soft.
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3
Quantification of Jn,c and Jy,c
The extended hard-sphere model involves use of the impulse that accounts for attractive impulse, Jy,c in the particle-wall model and Jn,c in the particle-particle
model. Physically, this impulse may be due to e.g. van der Waals forces or liquid bridging. In references [6, 7] Jy,c and Jn,c were, as mentioned, quantified in
two different ways in both cases, however, van der Waals interaction. Although
the approach is different between the two papers, both yield useful expressions
for the attractive impulse.
For particle-wall collisions [6] it was assumed that van der Waals forces
act during approach to and departure from the collision, while the duration of
contact was assumed short so that the attractive impulse during contact could
be neglected.
For particle-particle collisions the above method proved to be problematic,
and the opposite assumptions were made: the attractive interaction during approach and departure was neglected and the attractive impulse during contact
was estimated. The contact duration was determined using the analytical solution for purely elastic collisions (see e.g. [18]) based on the classical work by
Hertz [13].
As mentioned before, this second approach can also be applied to particlewall collisions, and the goal of the present paper is to improve on this method
by improving the estimation of the contact duration, taking into account also
attractive interaction during contact instead of only Hertzian theory.
Various approaches for modeling interaction between particles in the presence of attractive forces are available in the literature. Chronologically some
of the best-known models are the model of Derjaguin [19], the JKR-theory [14]
and the DMT-theory [20]. These papers inspired some other approaches that
offered better accuracy or easier implementation, e.g. [21–26]. The work in this
paper will be based on the JKR-theory, which is less complex than many other
theories and also leads to satisfactory results. In the following this theory is
briefly summarized for description of contact between two particles or a particle
and a wall. There is an extensive literature on this subject, some examples are
references [10, 14, 27].
Consider two spherical particles of radii r1 and r2 . The particles’ material
properties are described by their Young’s moduli and Poisson ratios: E1 , ν1
and E2 , ν2 respectively. An external load, FL , acting on the particles will
lead to their local deformation in the point of contact. The radius of this local
deformation is denoted as a and is a function of the particles’ material properties.
Hertz theory [13] describes this situation and the magnitude of a:
FL =
4 p
E∗ R∗ δ 3/2
3
and
δ=
a2
,
R∗
(18)
where the first equation relates the external load, FL , to δ, which is the sum of
the ’displacements’ of the surfaces in the contact point relative to the respective
particle centers, and the second equation relates δ to the contact radius, a. E∗
is the effective Young’s modulus, defined as: [(1 − ν12 )E1−1 + (1 − ν22 )E2−1 ]−1
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
and R∗ ≡ r1 r2 /(r1 + r2 ). Please note that this technique can be also used for
modelling particle-wall collisions: in this case one of the radii can be set to
infinity so that R∗ becomes r, the radius of the colliding particle.
The JKR theory accounts for an extra cohesive force acting between the
particles, one which augments FL , and causes a to be larger than that computed
from Hertz theory.
Rather than considering the adhesion directly, JKR analysis starts with the
assumption that the external load is higher than FL , F1 (> FL ), say. In this way
the extra cohesive interaction is mimicked by applying an appropriately higher
external load. Doing this, the contact radius, a, can be computed using the
classical work by Hertz but with F1 instead of FL :
F1 =
4 a3 E∗
3 R∗
and
δ1 =
a2
R∗
(19)
where δ1 is deformation due to the external load F1 .
The “elevated” load, F1 , is then decreased to the actual load FL , while at
the same time increasing the adhesion. During this process the contact area
between the particles remains constant, but the flattening, δ1 , decreases to δ,
the flattening corresponding to the external load FL . The JKR-theory then
indirectly makes an assumption that this corresponds to unloading of a flat
punch the radius of which is equal to a (see e.g. [11]) and this case can be
analytically described using the theory of elasticity: the relation between the
unloading force and the resulting change of deformation is namely:
F1 − FL = 2aE∗ (δ1 − δ).
(20)
Finally the surface energy per unit area, γ, that accounts for adhesion between the punch and the elastic particle is introduced. Omitting the details,
the final relation between the force Fadh = F1 − FL , which is the “extra” interaction force due to adhesion required for the present model, and the surface
energy becomes:
Fadh = (16πa3 E∗ γ)1/2 .
(21)
This also leads to the derivation of other expressions, such as the relation
between the indentation δ and contact radius a:
2 ao 1.5
a2
,
(22)
1−
δ=
R∗
3 a
with
ao =
9πγR∗2
E∗
1/3
.
(23)
Equation (21) is an expression for the adhesion force acting between the
particles (see also [15]). This force is a function of the contact radius and is
not constant during the collision process. The impulse due to this force, which,
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as mentioned in Introduction, is what we require for the extension of the hardsphere mode, can be now described as:
Z
Jn,c = (16πa3 E∗ γ)1/2 dt,
(24)
where only the contact radius, a, is a function of time. This is determined in
the following sections.
Determining the cohesive impulse requires analysis of the dynamics of the
particle impact using the soft-sphere method. The soft-sphere method was
pioneered by Cundall and Strack [16], and numerous other methods originated
from their classical work (e.g. [28] or [29], see also the review by Stevens and
Hrenya [17]).
While in the hard-sphere model the post-collisional velocities are obtained
as analytical expressions, the soft-sphere model is based on integration of the
particle equations of motion during the impact, which is computationally much
more expensive. The soft-sphere model makes it possible to account for cohesive
interaction during the impact and can therefore be used to quantify the cohesive
impulse for use in the hard-sphere model. This is the strategy adopted in the
following sections.
4
Dissipative force
The objective of this present work is thus to use the JKR-theory to account the
cohesive impulse.
A model recently proposed by Brilliantov et al. [15] is based on similar
strategy. This model was also compared with molecular dynamics simulations
for nanoparticle collisions in [30]. Other papers in the research literature also
attempt to estimate the dissipative forces acting during a collision. An example
is the recent paper by Marshall [31] that is based on earlier work by Tsuji et
al. [32] in which the dissipative forces are modelled making use of a damping
coefficient that is related to the restitution coefficient.
The paper by Brilliantov et al. [15], which is chosen as a basis for the present
work, derives the dissipative force in the following form:
!
r
3 6πγa da
3a2
−
,
(25)
Fdis = A
DR∗
2
D
dt
where D is 3/(4E∗ ). The parameter A is a function of viscous constants involved
in the modelling of the particles as viscoelastic bodies. In the following section
we assume this as input parameter.
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5
Numerical integration of the cohesive impulse
While the particles are in contact their collision can be described using the
following equation of motion:
m∗ δ̈ = −
4p
R∗ E∗ δ 3/2 − Fdis + Fadh ,
3
(26)
where the first term on the right-hand side accounts for the Hertzian force, the
second term for dissipation (see Eq. 25) and the third one for adhesion (Eq. 21).
The initial conditions can be described as:
and
δ(0) = 0
(27)
δ̇(0) = v0 = n · G(0) ,
(28)
which describes the relative velocity in the normal direction before contact.
The equation of motion (Eq. 26) was solved numerically, using the 4th order
Runge-Kutta scheme, in the following way: First a suitable time-step for the
numerical integration was determined based on a rough estimate for the duration
of contact using the analytical expression (see e.g [18]):
t̆col = 0.238
m2∗
E∗2 R∗ v0
0.2
.
(29)
This expression is for a collision without adhesion. It nevertheless makes it
possible to estimate a suitable time step for the numerical simulation chosen
here as: dt = t̆col /104 . Please note that this rough estimation will not lead
to erroneous results: for a case with appreciable adhesion the real duration of
contact will be longer than that computed from Eq. (28). The chosen time step
is therefore always small enough.
Solving Eq. (26) requires simultaneous calculation of the contact radius a
which is needed for finding e.g. the dissipative force Fdis . This was done by
solving Eq. (22) by the Newton-Raphson method.
In each computational step the integral represented by Eq. 24 was updated.
At the end, as the particles lose the contact, the deformation, δ, becomes,
according to the JKR-theory (see for example [10]):
δ(tcol ) = δc = −
1 a2o
.
3 42/3 R∗
(30)
This strategy also makes it possible to estimate the restitution coefficient,
which is also one of the inputs to the hard-sphere model, it can be calculated
as:
δ̇(tcol )
,
(31)
e=−
v0
where v0 is the normal component of the relative velocity before the collision,
equal to n · G(0) in the model for particle-particle collisions and vy in the model
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for particle-wall collisions. The minus sign accounts for the fact that the speed
at the end of the collision process becomes negative. Please note that e is
the traditional restitution coefficient based on the ratio of the outgoing to the
incoming normal velocities, taking into account all the interaction between the
colliding particles and is thus not the same as em used above. em can also
be found numerically by assuming no adhesion between particles, i.e. using
Eq. (26) with Fadh set equal to zero.
Please note that a similar technique to that described in this section could
be used for estimating tangential deformation. However, although the classical
soft-sphere model does envisage such a tangential deformation, the hard-sphere
model used in this work does not, it only operates with either Coulombian friction or rolling in the tangential direction. It would therefore not be correct to
consider deformation in the tangential direction in order to quantify the tangential impulse for the model used in this work. In the hard-sphere model used
in this work the particles, in case (i), can leave the collision with zero relative
surface velocity in the contact point, but not “spring back” from each other in
the tangential direction to e.g. leave the collision with a relative surface velocity
reversed to that with which they approached each other. However, there are
other hard-sphere models that do consider deformation in the tangential direction with the possibility that the surfaces of the colliding particle will “spring
back” from each other in the tangential direction.
6
Selected results
The objective of this section is to present some example calculation results for
the impulse Jn,c to illustrate the model described above. For these calculations
the following collision parameters, that were used and validated in [15] for the
collision of two ice crystals, are selected: particle density equal to 1000 kg/m3 ,
a Young’s modulus of 7.0 GPa and a Poisson ratio of 0.25. The simulations
were run for a range of values of surface energies (between 0.0 and 1.0 J/m2 )
to study the influence of adhesion. Two values of the viscoelastic parameter
A were used, namely 10−4 s and 10−5 s (note the units [15]) the second value
corresponding to a more elastic collision. Also two values of the effective radius
R∗ were used, namely 0.01 m and 0.02 m, and two initial relative speeds were
used, namely 0.002 and 0.05 m/s.
Figures 3 and 4 show the calculated attractive impulse, Jn,c , for various
cases. Increases in attractive impulse are observed as the surface energy and
particle diameter increase. The impulse is not finite if agglomeration occurs
(by definition it goes to infinity since the particles stay in contact) and this
was observed for small particles and high values for the surface energy, as well
as for low precollisional relative speeds. All these observations are in line with
expectation.
The influence of the viscoelastic parameter A is perhaps less intuitively obvious. Figure 4a shows the attractive impulse, Jn,c , for the lower of the two
pre-collisional relative normal speeds, 0.002 m/s. It can be seen here that at
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the higher values of surface energy the attractive impulse, Jn,c , is larger for the
higher value of A (10−4 s), i.e. for the more dissipative case. The reason is a
longer collision duration than for the more elastic case of A = 10−5 s.
However, when the surface energy is lower than about 0.35 J/m2 and for
the entire range of surface energies for the higher relative speed of 0.005 m/s
in Figure 4b this trend is reversed and Jn,c is larger for the lower value of
A. Here another effect dominates, which is illustrated in Fig. 5 showing the
collision duration and the maximum contact radius amax during collision as a
function of surface energy. These two graphs were obtained for the initial speed
0.002 m/s and R∗ = 0.02 m, i.e. they correspond to Fig. 4a. The figures show
that the more elastic collision results in an increase in the maximal contact
area. This is due to the the dissipative force being much lower and thus not
“braking” the approaching particles effectively. The higher contact area will
increase the attractive force and this effect dominates the opposing effect of the
higher collision duration under some conditions.
7
Dimensional analysis: estimation of the cohesive impulse
In the previous sections it has been shown how the cohesive impulse necessary
for the extended hard-sphere model can be calculated by analysis of collision
dynamics. This technique is, however, time-consuming and impractical. The
objective of this paper is to quantify the cohesive impulse for use in the hardsphere model obtaining analytical (rather than numerical) expressions for the
post-collisional properties of the particles.
In order to do this, dimensional analysis is used in this section to obtain an
analytical expression for the cohesive impulse that can be used directly in the
hard-sphere model.
The analyses in the previous sections show that the attractive impulse Jn,c
depends on the following parameters:
Jn,c = f (R∗ , m∗ , E∗ , γ, A, v0 ).
(32)
Using the Buckingham Π theorem [33] to arrange this relation in terms of dimensionless groups yields:
Jn,c
R∗ m∗ E∗ v0 A
.
(33)
=
,
,
v0 γA2
v0 A γA2
γ
A similar strategy is used in [34], where the goal is to estimate the value
of the damping coefficient (used frequently in discrete-element methods) as a
function of the restitution coefficient.
A functional form may be found by fitting a relation for the cohesive impulse involving these parameters to a series of numerical experiments using the
techniques described above. This was done varying the dimensionless groups in
the following ranges:
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a)
b)
Figure 3: The calculated cohesive impulse Jn,c for two initial relative speeds:
0.002 m/s (a) and 0.05 m/s (b), for R∗ = 0.01m as a function of surface energy.
The solid line corresponds to A = 10−5 s and the dashed line corresponds to
A = 10−4 s. The vertical thin line illustrates agglomeration that occurs for
A = 10−4 s.
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a)
b)
Figure 4: The calculated cohesive impulse Jn,c for two initial relative speeds:
0.002 m/s (a) and 0.05 m/s (b), for R∗ = 0.02m as a function of surface energy.
The solid line corresponds to A = 10−5 s and the dashed line corresponds to
A = 10−4 s.
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a)
b)
Figure 5: Collision duration (a) and the maximum contact radius (b) as a
function of surface energy, for relative speed 0.002 m/s and R∗ = 0.02m. The
solid line corresponds to A = 10−5 s and the dashed line corresponds to A =
10−4 s.
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• 1.67 · 105 < R∗ /(v0 A) < 11.67 · 105
• 5.4 · 1012 < m∗ /(γA2 ) < 2.76 · 1013
• 1.62 · 102 < E∗ v0 A/(γ) < 1.62 · 103 ,
The fitted relation is:
2
Jn,c = 269.5 · v0 γA
R∗
v0 A
0.3706 −0.2529
0.6172 m∗
E∗ v0 A
·
·
γA2
γ
(34)
Figure 6 shows a parity plot that compares Jn,c calculated by the fitted relation
above with those obtained from the numerical experiments.
Figure 6: The parity plot that validates π = Jn,c /(v0 γA2 ) calculated by using
the fitted function (Eq. 34) and by numerical experiments basing on Eq. 26
A similar strategy can be used for calculating the restitution coefficient em ,
i.e. following this approach it is no longer necessary to assume a value for this
parameter, since it can be calculated if the viscoelastic coefficient A is known.
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Conversely, if the restitution coefficient is known, the coefficient A can be also
determined using the model above.
The restitution coefficient is a function of the following parameters:
em = f (R∗ , m∗ , E∗ , A, v0 ),
(35)
where, as mentioned above, the surface energy γ is zero because em is defined
as the restitution coefficient resulting solely from the elastic properties of the
particles, neglecting any cohesion.
Arranging the variables in dimensionless groups gives:
R∗ E∗ A2 v0 A
em =
.
(36)
,
m∗
R∗
By running a series of numerical experiments using model Eq. 34 with the
cohesive force set to zero, and fitting the results to a suitable functional form
gives:
−0.007465
v0 A
R∗ E∗ A2
(37)
·
em = 0.8957 exp −0.7206 ·
m∗
R∗
where the dimensionless groups were varied within the ranges:
• 0.005 <
R∗ E ∗ A 2
m∗
• 1.0 · 10−6 <
8
v0 A
R∗
< 0.05
< 1.9 · 10−5
Selected results of particle-particle collision
modelling
In this section some results are shown where the extended hard-sphere model
has been used. Three examples are shown: a collision between two particles, a
collision between two particle clouds and a laminar pipe flow laden with particles. The objective is not to study the physics of these processes, but only to
illustrate the use of the extended hard-sphere model.
8.1
Collisions of two particles
First the collision of two particles that move along the x-axis with a speed Uinit
as shown in Fig. 7 is investigated. If the collision does not lead to agglomeration,
each particle will obtain a new velocity that can be calculated using the relations
given above. The absolute value of the linear post-collisional velocity is denoted
by Uend .
The two particles are identical, the particle density is ρ = 1000 kg/m3 ,
Young’s modulus is E = 7 GPa, the Poisson coefficient is ν = 0.25, the friction
coefficient is µ = 0.15 and the viscoelastic coefficient A is 10−6 s. The particle
radius is taken as 5·10−3 m, which corresponds to R∗ = 2.5·10−3 m.
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Figure 7: Illustration of the case investigated in this section: two identical
particles move along x-axis toward each other.
Figure 8 shows the relation between the initial, Uinit , and the final speed,
Uend , of the particles (please note that in this selected case both particles obtain
the same post-collisional velocity magnitude). Also Fig. 9 shows the relation
between the initial speed and the final angular velocity of the particles. Calculations were carried out for two values of the surface energy: γ = 0 and γ =
0.37 J/m2 .
For the case when the surface energy was 0.37 J/m2 and the initial speed was
low (about 0.047 m/s), the particles do not bounce off but form an agglomerate.
The final linear velocity of the agglomerate is in this situation equal to zero.
The extended model reduces to the standard hard-sphere model for γ =
0.0 J/m2, and this is also indicated in the figure. All real particles will have
some cohesion acting between them, so that the classical hard-sphere model can
be seen as a limiting case.
8.2
Collisions of two particle clouds
Another application of the model is to study a collision between two clouds of
particles moving towards each other as shown in Fig. 10. In this simulation it
is assumed that there is no fluid present in the system, so that the particles
lose their kinetic energy only due to dissipative processes in the collisions. The
collision algorithm was the same as in previous papers (see [35, 36]), but using
the new quantification of Jn,c .
This set-up may model cosmic dusts in e.g. Saturn’s rings. For these simulations, therefore, the same physical parameters were used as in [15], where the
objective was similar. These parameters are ρ: 1000 kg/m3 , E: 7 GPa, ν: 0.25
and γ: 0.37 J/m2 (ice particles). A was assumed as 10−6 s, and the particle radius was set to 5·10−3 m (or R∗ = 2.5·10−3). In addition, the Coulombic friction
coefficient was set to 0.05 (selected as a “reasonable value”, since this coefficient
depends on many factors, such as the temperature, the height of asperities on
the particle surface, etc). The two clouds were assumed to move towards each
other with an initial speed 0.05 m/s.
The total number of particles was 8192 (4096 particles in each cloud). The
size of the clouds was 0.853 m in the x-direction and 0.64 m in the y- and zdirections. The individual particles were distributed randomly. Agglomerates
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Figure 8: The relation between the final and initial velocity magnitude for each
particle for various values of surface energy, γ.
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Figure 9: The relation between the final angular velocity and linear initial
velocity magnitude for each particle for various values of surface energy, γ.
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(if formed) were modelled as spherical particles of a volume equal to the sum of
the volumes of the agglomerating particles.
Figure 10: Illustration of the collision of two particle clouds
Two snapshots of particle position at two points in time are shown in Fig. 11.
The snapshots are shown in the x-z plane, where the details are clearly seen.
Figure 11a shows particle positions after 10 seconds of simulation, where the
first collisions have been detected and one can observe a higher concentration
of particles in the zone where the clouds have inter penetrated. Figure 11b
corresponds to 28 seconds of simulation, at which point the moving clouds have
passed through each other. Some of the particles have, however, collided and
lost some of their kinetic energy.
One interesting aspect of the process can be analyzed by counting collisions,
distinguishing collisions leading to agglomerate formation. Figure 12 shows the
‘collision efficiency’, defined as the number of collisions leading agglomeration
divided by the total number of collisions, plotted against the surface energy
of the particles, the surface energies taking on values of 0.37 (which is the
case in the snapshots in Fig. 11), 0.185 and 0.092 J/m2 (the last two values
are respectively two and four times lower than the first one, these values were
selected solely to show the effect of cohesion). This plot corresponds to the
point in time equal to 20 s.
Figure 13 shows the relation between the average restitution coefficient (calculated for the first 200 collisions) and the surface energy. Please note that this
is the mechanical restitution coefficient, em , not the classical one defined as the
ratio of normal relative velocities after and before the collision. Since em is the
mechanical restitution coefficient, and therefore does not account for the effect
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Physical Review E 84, 031303, doi:10.1103/PhysRevE.84.031303
a)
b)
Figure 11: Two snapshots of particle position shown in the x-z plane after (a)
10 seconds; (b) 28 seconds.
of cohesion, it is not a direct function of the surface energy. However, as the
surface energy increases, the cohesive particle-particle interactions leads to more
“damping” in the system causing the particles to collide with a lower relative
velocities. This, in turn, leads to higher values of the mechanical restitution
coefficient (em , Eq. 37). The simulation results shown in Fig. 13 thus supports
the validity of the simulation code.
8.3
Fluid flow with particles
In this section the following case is considered: a laminar incompressible flow
of a fluid laden with solid particles in a rectangular channel.
The height and the width of the channel were 0.64 m, while the length was
0.256 m. The boundary conditions at the two ends were periodic and no-slip
for the upper and lower walls. The average speed of the fluid was 0.015625
m/s, the kinematic viscosity was 10−5 m2 /s and fluid density was 1.0 kg/m3 .
Using the channel height as the reference length these parameters correspond
to a Reynolds number equal to 1000. The Coulombian friction coefficient was,
as in the previous example, set equal to 0.05.
During the start-up phase computational fluid dynamics simulations of the
flow of the pure carrier fluid were run for 2000 seconds at which time the flow
was assumed to have established, i.e. fluid flow velocity profile became almost
parabolic in the mid-part of the channel. Subsequently 2048 solid, spherical
particles were introduced into the channel at random positions.
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Figure 12: Collision efficiency for interaction between two colliding particle
clouds as a function of surface energy
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Figure 13: Average mechanical restitution coefficient, em , (for the first 200
collisions) as a function of particle surface energy
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The simulation code, the mathematical model and the numerical scheme
solving the flow were the same as in references [36]. Regarding the fluid flow,
the computational domain was discretized using a staggered grid. The numerical scheme was based on the Chorin projection [37], modified by use of the
Adams-Bashfort time-advancement scheme. For discretization of the convective
terms the Variable-Order Non-Oscillatory Scheme [38] was used. The code was
validated against benchmarks and grid-independence was assured.
The particles were modelled by writing and solving Newton’s second law
for each particle (Lagrangian particle tracking) where the main fluid-particle
interaction was considered to be the drag force, as shown in [36]. Particleparticle and particle-wall collisions were modelled using the algorithm shown
in [35]. The only modification in the algorithm is the implementation of the extended hard-sphere model, such that the post-collisional particle velocities were
different from those in simulations that uses the classical hard-sphere model.
In these simulations the new model was implemented only for particle-particle
collisions, while particle-wall collisions were modelled using the standard hardsphere model with restitution coefficient equal to 1.0 (i.e. these simulations
focused only on particle-particle collisions).
The results of the first case are shown in Fig. 14 where six simulation results,
where γ varied between 0.0 and 0.037 J/m2 ) are compared in terms of the
collision efficiency. The results shown are for a time of 3000 seconds after the
particles were introduced. The obtained collision efficiency is high because the
relative velocity between the particles was quite low in these simulations.
9
Concluding remarks
This paper focuses on the extended hard-sphere model where cohesive interaction is included, with as the main objective to quantify the cohesive impulse.
This impulse was determined by combining the soft-sphere approach and dimensional analysis.
The extended hard-sphere model can be used for Lagrangian simulations
of particle flows wherein the cohesive interactions cannot be neglected. Such
flows may be flows wherein the particles move with a low velocity relative to
each other and/or flows involving small particles, such as nanoparticles. Other
systems where the cohesive particle interaction is relevant are systems where
the particles are sticky or coated with some liquid giving rise to the formation
of liquid bridges between them. For these latter system, however, the cohesive
impulse still needs to be quantified; the present paper assumes only van der
Waals interaction.
The new technique can be also used for determination of collision efficiencies,
as illustrated for e.g. simulations of fluid flow laden with particles.
In this research, the model by Brillantov et al. [15] was used to estimate the
dissipative force acting during collision. A disadvantage of using this model for
quantification of the cohesive impulse is that the parameter A that may not be
directly known.
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Figure 14: Collision efficiency for interaction between particles in a laminar flow
as a function of surface energy
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Acknowledgements
The authors are grateful to Dr Boris V. Balakin for useful discussion regarding
the collision efficiency as well as testing of our CFD code. Funding for this work
for the Norwegian Research Council under the FRINAT program is gratefully
acknowledged.
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